APPLICATIONS OF FORWARD AND
FUTURES
1
Week 3
A RECAP OF FORWARD AND FUTURES
Contracts
Default Risk
Margin Requirements
Regulatory Requirements
Transaction Data
Application/Preferred Risk
Management Vehicle
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Forwards
Custom; OTC traded
Futures
Standardized; exchange
traded
Counterparty default risk
Clearinghouse guarantees
against default.
Contract value generally paid at Margin requirements and
expiration. Margins and interim daily mask-to-market to
settlements are occasionally
settle gains/losses.
used to mitigate counterparty
default risk
Essentially unregulated.
Regulated in U.S. at the
federal level
Generally unavailable to the
Reported to exchange and
public as transactions are
regulatory agencies
private
Interest rate resets on loans.
Bond and equity
Foreign currency risk
portfolios
Dealer transactions to
offset Eurodollar
swap/option interest rate
trades
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MOTIVATION

Why hedge?
 Companies can then focus on their main activities, for
which presumably they do have particular skills and
expertise.
 By hedging, they avoid unpleasant surprises such as
sharp rises in the price of a commodity that is being
purchased.

What are the arguments against hedging?
 Shareholders are usually well diversified and can make
their own hedging decisions
 It may increase risk to hedge when competitors do not
 Explaining a situation where there is a loss on the hedge
and a gain on the underlying can be difficult
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QUESTIONS TO BE ANSWERED
When is a short futures position appropriates？
 When is a long futures position appropriate？
 Which futures contract should be used？
 What is the optimal size of the futures position for reducing
risk？


Mode of hedging
 Hedge-and forget
 Futures contracts versus forward contracts
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LONG HEDGE, SHORT HEDGE, AND BASIS
 f1
: Initial Futures Price
 f2 : Final Futures Price
 S2 : Final Asset Price
 Hedge the future purchase of an asset by entering into a
long futures contract
 Cost of Asset=S2 – (f2 – f1) = f1 + Basis2
Hedge the future sale of an asset by entering into a short
futures contract
 Price Realized =S2 + (f1 – f2) = f1 + Basis2


Basis risk arises because of the uncertainty about the basis
when the hedge is closed out.
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MEASURING EQUITY RISK

Equity portfolio risk: beta

What is the beta of an equity index (proxy for market
portfolio) chosen as the performance benchmark?

How is the beta of an equity fund measured?
Risk level
p 
C(rp , rm )
Measure of the direction and
extent by which the equity
portfolio and the index move
together
 m2
Variance of the return on market
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MANAGING PORTFOLIO BETA

Dollar beta of the portfolio
market value of the portfolio  p  P  p
To modify the risk level of a portfolio, the target beta *
should be set in such a way that the sum of the dollar betas
of the existing portfolio and a specified number of futures
contract Nf.
 Dollar target beta
dollar beta of the portfolio + Nf  dollar beta of 1 futures contract

* P = p P + Nf f f m

Number of contracts required
  *  p   P 


Nf 
 f   f m


 
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multiplier
Futures price
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HEDGING EXAMPLE
S&P 500 cash value 1,000
 S&P 500 futures price is 1,010
 Value of Portfolio is $5,050,000
 Beta of portfolio is 1.5
 Beta of futures is 1.0
 Multiplier of S&P 500 is 250
 Risk-free interest rate = 4%
 Dividend yield on index = 1%


What position in futures contracts on the S&P 500 is
necessary to hedge the portfolio (i.e., * = 0) ?
1.5 5,050,000
Nf  

 30
1.0 2501,010
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EXAMPLE: ADJUSTING PORTFOLIO BETA

A manager of a $5,000,000 portfolio wants to increase the
beta from the current of 0.8 to 1.1. The beta on the futures
contract is 1.05, and the total futures price is $240,000.

What is the required number of futures contracts to achieve
a beta of 1.1?
1.1  0.8  $5,000,000
N f  
  5.95

 1.05  $240,000 

The appropriate strategy would be to take long position in 6
futures contracts. Taking a long position in index futures
contracts will increase the beta and leverage up the position.
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THE HEDGE ISN’T PERFECT (1)

Rounding to the nearest whole contract gives rise to imperfect hedge.

If the reference index (used to calculate the betas) increased in value by
2%; the value of the equity position increased by 1.6%; and the value of
the futures price increased by 2.1 %. These values correspond exactly to
what we would expect with the provided betas of 0.8 and 1.05.

Had the leveraged position worked as desired (i.e., achieved an effective
beta of 1.1), the value of the portfolio would have increased 1.1(0.02) =
2.2% to $5,110,000 = $5,000,000(1 + 0.022)

In our example, where we leveraged up the beta with 6 contracts, the
profit from the futures contract position is
$30,240 = 6($240,000)(1.021)- 6($240,000)

The profit from the equity position itself is
$80,000 = $5,000,000(1.016) - $5,000,000

Therefore, the final value of the equity portfolio plus futures position is:
$5,110,240 = $30,240 + $80,000 + $5,000,000
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THE HEDGE ISN’T PERFECT (2)

The return on the position is
0.022048 = ($30,240 + $80,000) / $5,000,000

The effective beta on the portfolio proved to be
effec tivebeta 
% c hangeof portfolio value 0.022048

 1.1024
% c hangeof index value
0.020

In this case, the discrepancy was due to rounding in the
futures position.

There is often other error from the fact that the portfolio and
futures contracts are not perfectly correlated with the index.
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CLASS EXERCISE

A fund has $400 million invested in a diversified portfolio of
common stocks with a beta of 1.1 relative to S&P 500, which
is at the level of 968.00. The six-month S&P 500 futures is
trading at 998.00 and has a beta of 0.95. The dollar multiplier
for the futures contract is 250.
1.
To reduce the beta of the portfolio to 0.90, how many
contracts should the fund manager trade (buy or sell)?
2.
The S&P 500 index is down 3% at the futures expiration
date. The portfolio is down 3.4%; the futures are trading at
969.56. What is the value of the overall position (stock
portfolio + futures) and the effective beta?
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ADVANTAGES OF FUTURES

Lower Transaction Costs – The use of futures contracts is
preferable to liquidation/purchase of portfolio securities,
especially over short time horizon.

Asset Allocation Revisions – Futures allow managers to
make asset allocation changes without disturbing the
underlying portfolio.

Transaction Time – Risk management strategies can be
quickly initiated in response to a particular forecast, and
quickly closed out should the outlook change.

Contract Liquidity – Futures contracts are generally more
liquid than a portfolio’s individual securities.
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DISADVANTAGES

Potentially Divergent Risk Management Outcomes – Betas can be unstable
and are hard to measure. If a portfolio’s beta value does not capture the
portfolio’s actual sensitivity to underlying sources of risk, then the hedging
process will be inexact. As a result, the actual outcome may diverge from the
desired outcome.

Contract Liquidity – Futures are not exempt from liquidity problems. The
liquidity of long-term futures is significantly lower than that of short-term
futures.

Entity Liquidity Needs – Although they require less capital to trade, futures
cannot solve an investor’s immediate liquidity needs. For example, to meet
an impending cash requirement, underlying securities still have to be
liquidated.

Leverage – Futures positions are leveraged; losses (or foregone gains) on
purchases or sales of futures are potentially large in percentages. For this
reason, some firms prohibit the use of these instruments.
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MANAGED FUTURES

“Managed futures” is a diverse collection of active hedge fund
trading strategies that specialize in liquid, transparent, exchangetraded futures markets and deep foreign exchange markets

Managed futures traders are commonly referred to as
“Commodity Trading Advisors” or “CTAs,” a designation which
refers to a manager’s registration status with the Commodity
Futures Trading Commission and National Futures Association.

Most CTAs trade equity index, fixed income, and foreign
exchange futures. They don’t simply take on systematic exposure
to an asset class, or beta, but are attempting to add alpha through
active management and the freedom to enter short or spread
positions, which can result in totally different return profiles than
the long-only passive indices.
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MANAGED FUTURES GROWTH IN ASSETS UNDER
MANAGEMENT 1980-2008
AUM ($billions)
Source: AlphaMetrix Alternative Investment Advisors, BarclayHedge Alternative Investment Database
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BARCLAYHEDGE’S BTOP50 INDEX
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
Dec01987
Feb 50
1991
Apr100
1995
Jun150
1999
Source: BarclayHedge Alternative Investment Database
Aug200
2003
2502007
Oct
300 2011
Dec
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PERFORMANCE OF MANAGED FUTURES

Maximum, Minimum, and Mean Rolling Return of Barclays
Capital BTOP 50 Index Over Different Holding Periods (Jan
1987 through Dec 2008)
Rolling Rate of Return
Months
Source: AlphaMetrix Alternative Investment Advisors, Bloomberg
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CASE STUDY
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OPTIMAL HEDGING WITH INDEX FUTURES (1)


An institutional investor holds a portfolio of Japanese stocks
that has returns following closely that of the Nikkei 225
stock index returns St+1/St , where St+1 = St+1 – St .
The value of the portfolio in yens is NS  St, i.e., NS shares of
a “stock” called Nikkei 225.

Multiplier of Nikkei 225 futures traded on SGX is ¥500.

Value of the hedged portfolio with Nf futures contracts is
Vt = NS  St – Nf  500  ft  P – F
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OPTIMAL HEDGING WITH INDEX FUTURES (2)


The change in portfolio value is
Vt+1 = NS  St+1 – Nf  500   ft+1  P – F
The investor minimizes the risk or variance of Vt+1 by
finding an optimal number Nf such that the risk is minimum.
VVt 1   NS2 VSt 1   N 2f  5002 Vft 1   2NS  N f  500 CSt 1 , ft 1 

Solving the first order condition with respect to Nf leads to
NSCSt 1 , ft 1 
N 
500Vft 1 
*
f
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OPTIMAL HEDGE RATIO
o
o
o
S : standard deviation of S, the change in the spot
price during the hedging period
f : standard deviation of f, the change in the
futures price during the hedging period
r : the coefficient of correlation between S and f
NSCSt 1 , ft 1  NS rS f NS  S
N 


r
2
500Vft 1 
500  f
500  f
*
f
o
h*: hedge ratio that minimizes the variance of the
hedger’s total position in the underlying and the
futures.
500 N *f

h* 
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NS
r
S
f
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MINIMUM VARIANCE HEDGE

To hedge the risk of an index portfolio, the number of
contracts that should be shorted is
 S
 S
f 
f 
N S St f t C t 1 , t 1  N S St C t 1 , t 1 
St
ft 
St
ft 
N S CSt 1 , f t 1 
P


*
Nf 



500V f t 1 
500 f t
 f 
 f 
500 f t 2 V t 1 
500 f t V t 1 
 ft 
 ft 
P
N 
fm
*
f

Beta of the index futures can be estimated with the following
 St 1 ft 1 
linear specification


f
St 1
    t 1  e t 1
St
ft
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C
,
St
ft 


 f t  1 

V 
 ft 
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CASE STUDY (1)
An asset management company has a long position of 500
lots in STI ETF. The volume-weighted average price per
share is S$3.4567 when the STI ETF shares were purchased.
 Using a proprietary quant strategy, the fund holds the view
that over the next 2 weeks, STI is expected to decline.
 To protect the fund’s value, the fund decides to use a futures
contract to hedge.
 Naturally, the fund looks into STI futures traded on SGX.


Question: What is the book value of the long position in STI
ETF?

Question: Should the fund perform a long hedge or a short
hedge?
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CASE STUDY (2)

Unfortunately, the liquidity of STI futures is almost zero.
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CASE STUDY (3)

By contrast, futures on MSCI Singapore index are more
liquid.
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CASE STUDY (4)

Suppose the said 2 weeks start from November 9, 2010.

Question: Which maturity month of SiMSCI should the fund
use?

Question: How many contracts should the fund cross-hedge
with SiMSCI futures to achieve an optimal result?
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